For a Fan and Lights I would consider 14 AWG to be the absolute minimum, personally I would use 10 AWG
I hate voltage drop, its a waste of power and causes waste heat.
Overdoing the wire costs next to nothing in weight or money and saves you power in every use. (I dont really consider it as Overdoing it, to me its doing the Job right)
BTW sound (twin) wiring underates the wire even further 16-2 is under size for a fan and lights, this underating is further exacerbated by the fact that the wire is buried